As other's have mentioned - Yes you can use your miles to buy tickets for other people - even friends who are of no relation.
I've used my miles to help out friends and get them a ticket plenty of times since I rarely use the miles on myself (always chasing status).
And as LBJ mentioned, if you're kids are flying on a redemption ticket - they will not earn miles for those flights. If they already have earned some miles and you want to ensure they don't expire - you need to ensure they have some mileage activity at least once every 24 months. The easiest way to do that would be to try to make sure you take them on a revenue (not redeemed) flight at least once every 2 years if possible.
If you're coming up on 2 years of inactivity - you might be able to talk to customer service and let them know they're minors and see if they can grant an extension - not guaranteed but the customer service is generally pretty nice with Alaska.